2.4: Quadratic Inequalities
- Page ID
- 233
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Solving Quadratic Equations
We solve quadratic equations by either factoring or using the quadratic formula.
Definition: The Discriminant
We define the discriminant of the quadratic
\[ax^2 + bx + c \]
as
\[D = b^2 - 4ac.\]
The discriminant is the number under the square root in the quadratic formula. We immediately get
D | # of Roots |
> 0 | 2 |
< 0 | 0 |
0 | 1 |
so that the quadratic has no real roots.
Quadratic Inequalities
\[x^2- x - 6 > 0\]
Solution:
First we solve the equality by factoring:
\[(x - 3)(x + 2) = 0\]
hence
\[x = -2 \; \text{ or } \; x = 3.\]
Next we cut the number line into three regions:
\[x < -2, -2 < x < 3, \text{ and } x > 3.\]
On the first region (test \(x = -3\)), the quadratic is positive, on the second region (test \(x = 0\)) the quadratic is negative, and on the third region (test \(x = 5\)) the quadratic is positive.
Region | Test Value | y-Value | Sign |
---|---|---|---|
\(x < 2\) | \(x = -3\) | \(y = 6\) | \(+\) |
\(-2 < x < 3\) | \(x = 0\) | \(y = -6\) | \(-\) |
\(x > 3\) | \(x = 5\) | \(y = 14\) | \(+\) |
We are after the positive values since the equation is "\(> 0\)". Hence our solution is region 1 and region 2:
\[x < -2 \; \text{ or } \; x > 3.\]
We will see how to verify this on a graphing calculator by noticing that
\[y = x^2 - x - 6 \]
stays above the x-axis when \(x < -2\) and when \(x > 3\).
Applications
since -.1 does not make sense, we can say that the radius of the garden is 1.1 feet.
Where \(x\) represents the number of skiers that come on a given day. How many skiers paying for Heavenly will produce the maximal profit?
Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.